We study Higgs boson production in association with a top quark and a W boson at the LHC. At NLO in QCD, tWH interferes with $t\bar{t} H$ and a procedure to meaningfully separate the two processes needs to be employed. In order to define tWH production for both total rates and differential distributions, we consider the diagram removal and diagram subtraction techniques that have been previously proposed for treating intermediate resonances at NLO, in particular in the context of tW production. These techniques feature approximations that need to be carefully taken into account when theoretical predictions are compared to experimental measurements. To this aim, we first critically revisit the tW process, for which an extensive literature exists and where an analogous interference with $t \bar{t}$ production takes place. We then provide robust results for total and differential cross sections for tW and tWH at 13 TeV, also matching short-distance events to a parton shower. We formulate a reliable prescription to estimate the theoretical uncertainties, including those associated to the very definition of the process at NLO. Finally, we study the sensitivity to a non-Standard-Model relative phase between the Higgs couplings to the top quark and to the W boson in tWH production.
